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Abstract:We introduce the definition of $m$-to-$1$ mappings between two finite sets, which unifies and generalizes the definitions of $2$-to-$1$ and $n$-to-$1$ mappings in recent literature. We also characterize these $m$-to-$1$ mappings in terms of the generalized local criterion and thus provide three generic constructions of $m$-to-$1$ mappings, which unify and generalize the previous known constructions. Using these constructions, the problem whether $x^r h(x^s)$ is $m$-to-$1$ on the multiplicative group $\mathbb{F}_{q}^{*}$ is converted into that whether an associated polynomial $x^{r_1} h(x)^{s_1}$ is $m_2$-to-$1$ on the order~$\ell$ subgroup~$U_{\ell}$ of $\mathbb{F}_{q}^{*}$, where $m_2 = m / (r, s)$ and $\ell = (q-1) / s$. Furthermore, the $m_2$-to-$1$ property of $x^{r_1} h(x)^{s_1}$ on $U_{\ell}$ is studied in detail in four different cases. In addition, a recursive construction of $m$-to-$1$ mappings from $m$-to-$1$ mappings is proposed.

Submission history

From: Yanbin Zheng [view email]
[v1] Thu, 8 Aug 2024 05:05:42 UTC (91 KB)
[v2] Sat, 8 Aug 2026 15:26:02 UTC (74 KB)